Here, E(Re) means the isophote which encloses
half of the total light, and
E(Re) is the
isophote at a radius
(0 < < 1) times
Re. This is a flux ratio.
For a Sérsic profile which extends to infinity,

(21)

This expression is a monotonically increasing function of n, and
for = 1/3 it's values are
shown in Figure (3).
An often unrealized point is that if every elliptical galaxy had an
R1/4 profile then they would all have exactly the same
degree of concentration. Observational errors aside, it is only because
elliptical galaxy light-profiles deviate from the R1/4
model that a range of concentrations exist. This is true for all
objective concentration indices in use today.

Figure 3. The central concentration
CRe(1/3), as defined by
Trujillo et al. (2001c),
is a monotonically increasing function of the Sérsic index
n.

It should be noted that astronomers don't actually know where the
edges of elliptical galaxies occur; their light-profiles appear to
peeter-out into the background noise of the sky. Due to the presence
of faint, high-redshift galaxies and scattered light, it is not
possible to determine the sky-background to an infinite degree of
accuracy. From an analysis of such sky-background noise sources,
Dalcanton & Bernstein
(2000,
see also
Capaccioli & de
Vaucouleurs 1983)
determined the limiting surface brightness threshold to be
~ 29.5 B-mag arcsec-2 and
µ ~ 29 R-mag arcsec-2. Such depths are
practically never achieved and shallow exposures often fail to include a
significant fraction of an elliptical galaxy's light. One would
therefore like to know how the concentration index may vary when
different galaxy radial extents are sampled but no effort is made to
account for the missed galaxy flux. The resultant impact on
CRe and other popular concentration indices
is addressed in
Graham, Trujillo, &
Caon (2001).

It was actually, in part, because of the unstable nature of the
popular concentration indices that
Trujillo et al. (2001c)
introduced the notably more stable index given in Equations (20) and (21).
The other reason was because the concentration index
C() =
i,j
E()Iij /
i,j
E(1)Iij,
where E()
denotes some inner radius which is
(0 <
< 1)
times the outermost radius which has been normalized to 1
(Okamura, Kodaira, &
Watanabe 1984;
Doi, Fukugita, &
Okamura 1993;
Abraham et al. 1994),
should tend to 1 for practically every profile that is sampled to
a large enough radius. It is only because of measurement errors,
undersampling, or the presence of truncated profiles such as the
exponential disks in spiral galaxies, that this index deviates from a
value of 1.